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Designing Frequency-Dependent Impedances?

Discussion in 'Electronic Design' started by Diego Stutzer, Feb 19, 2004.

  1. Hi,
    Every one knows, that e.g. a simple RC-parallel circuit has a
    frequency-dependent impedance-characteristic (Absolute Value) - the
    impedance (Abs) raises as the Frequency approaches zero. As a formula: Zin
    = 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

    Now the hard part. How does one create an Impedance, which decreases
    "slower", for frequencies close to zero but then decreases "faster" for
    higher frequencies, than the simple parallel RC-Circuit?
    Is there some kind of procedure like the one for syntesizeing LC-Filters
    (Butterworth, Chebychev,..)?

    Simply increasing C does not really help, because this equals a factoring of
    the frequency.
    Increasing R does not help as well, as it seems.


    I hope one of you cracks can help me out.
    So far, thanks for reading.
    Diego Stutzer
     
  2. Cecil Moore

    Cecil Moore Guest

    For emulation modeling, there exist programmable resistors,
    capacitors, and inductors. Is that what you have in mind?
     
  3. The frequency behaviour of R, L and C corrsponds to differential
    equations with one or the other value as parameter. Meaning
    you're not completetly free. A book on filter design may help
    clear some gaps up.
    Basically : you can build what can be approximated as rational
    polynominal in frequency space.

    Rene
     
  4. John Jardine

    John Jardine Guest

    You need to graph out the required frequency-impedance slope then
    approximate the required roll off rates using a segmented breakpoint scheme
    consisting of a number of CR series sections in parallel. Essentially it's a
    straight line approximation to the required Z-F curve. The CR's adding
    zeroes as the frequency goes up.

    Estimating the individual time constants can be irksome as each has effect
    outside it's area of interest. Use a 'least-squares approximation' to obtain
    a best curve fit for the number of sections involved.

    It's an interesting subject but I've come across nothing out there that's of
    use.

    regards
    john
     
  5. Steve Nosko

    Steve Nosko Guest

    In other words YES. You use combinations of resistors and capacitors or
    inductors. Understanding the concept of "poles" and "Zeroes" is one way
    which allows the synthesis of such circuits. Another is the concept of
    "corner Frequency".
     
  6. Steve Nosko

    Steve Nosko Guest

    I think he is looking for slopes of less that 6 dB per octave.
     
  7. Ban

    Ban Guest

    ___
    o-|___|--+--------+--------+---o
    10k | | |
    | | |
    --- --- ---
    --- --- ---
    |100n |10n | 3n3
    .-. .-. |
    | | | | |
    | |15k | |10k |
    '-' '-' |
    | | |
    o--------+--------+--------+---o
    created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de
    use fixed font to view

    This does exactly what you want, in the beginning the slope is less than
    3dB/oct. and at 10kHz it goes to 6dB/oct.
    This is how to produce a pink noise that rolls off faster at the end of
    range, or to make some weighted filters (dBA) etc.

    ciao Ban
     
  8. Guy Macon

    Guy Macon Guest

    That's a great resource! See:
    http://www.tech-chat.de/AAcircuit.html
    http://www.tech-chat.de/aacircuit_tutorial.htm

    Is there a human-generated english translation? If not,
    I can do a machine translation with the usual humorous
    but usable results.
     
  9. John Jardine

    John Jardine Guest

    [Slightly OT].These 'spread CR' things are *weird*. How else can 1Hz to 1MHz
    be set with just one pot!.

    ,-------------------+--------------.
    | | |
    .-. .-. |
    2k7| | | | |
    | | | | |
    '-' '-'220 |
    | | |
    | ¦ V+ |
    | | |\| |
    ,---+---+--++---+---+--------- | ------|-\ |
    | | | | | | Min.-. | >---'-o
    .-. .-. .-. .-. .-. .-. | |<-----|+/ Square wave out
    | | | | | | | | | | | | | | |/|
    | | | | | | | | | | | | Max'-'Pot V-
    '-' '-' '-' '-' '-' '-' | 10k
    | | | | | | |
    --- --- --- --- --- --- Comparitor
    --- --- --- --- --- --- |
    A| B| C| D| E| F| .-.
    '---+---+---+---+---' | |680
    | | |
    0V '-'
    A=10K:10u |
    B=4k7:1u 0V
    C=2k2:100n
    D=1k2:100n
    E=680:1n
    F=330:100p

    created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

    regards
    john
     
  10. Hi Diego,

    This can be done in a number of ways employing active components (I've
    just seen the drift of the thread take on this complexity).

    The Bi-Quad filter comes to mind and has been around implemented with
    Op-Amps for quite a while.

    Another is the cascaded, bucket-brigade chip from reticon (haven't
    played with this for 20 years tho').

    You could also build an MCU interfacing ADC and DAC chips or simply
    step up to DSP chips for filters that are impossible to implement in
    any combination of passive L-C-R combinations.

    However, the Bi-Quad offers simultaneous Low Pass, Band Pass, Band
    Reject, and High Pass outputs from one circuit configuration. For
    playing around with, that flexibility is hard to beat.

    73's
    Richard Clark, KB7QHC
     
  11. Max Hauser

    Max Hauser Guest

    "Diego Stutzer" in news:...
    What you are asking about is a form of what's traditionally called the
    network synthesis problem (creating a network of components to realize a
    prescribed signal response) and specifically the synthesis of a one-port, or
    impedance.

    At one time (when phone companies ruled the earth and computers had
    conquered few signals and DSP was reserved for BIG things like the US
    Perimeter Acquisition Radar at Concrete, North Dakota -- affectionately the
    "PAR"), this was a popular subject in engineering schools at the
    advanced-undergrad or graduate level. It is still extremely important
    sometimes, especially with the sophisticated signal processing used today on
    continuous-time signals in consumer products. A host of
    applied-mathematical techniques (Foster and Cauer synthesis, Brune's
    impedance-synthesis lemma, etc.) apply even to one-ports. Some of them are
    highly counterintuitive. Not, in other words, a subject perfectly matched
    to the contraints of brief advice on newsgroups. (Note also that
    Butterworth and Chebyshev approximants are mathematical methods to approach
    one group of curves out of things that naturally give you a different type
    of characteristic -- "Butterworth and Chebyshev" have nothing to do with
    specific circuit topologies or components). If you want to pursue it
    further I could suggest investigating "network synthesis." Temes and
    LaPatra had a reasonable modern (1970s) book about it. Karl Willy Wagner
    started it all in 1915 by inventing filters.

    Richard Clark suggested also investigating the small op-amp "biquad"
    networks for designable frequency response (actually you can turn them into
    one-ports, the so-called shunt-filter class, but again a bit of a subject
    for a brief response). Note that technically a "bi-quad" is any network
    giving a biquadratic transfer function (2nd-order numerator and denominator)
    though in RC-active filters it's often applied to the closely related
    Åkerberg-Mossberg and Tow-Thomas configurations. For practical info see van
    Valkenberg's excellent general introductory book on filters from the 1980s.

    For an accessible modern example of these small op-amp-based "biquad"
    networks, look up the LTC1562 from Linear Technology, a commercial chip with
    four trimmed "biquad" networks, programmable by outboard components for
    applications from a few kHz to a few hundred kHz.
     
  12. Diego:

    You cannot do *exactly* what you propose, but you can get arbitrarily
    close to it.

    The "closeness" being a function of the cost you are prepared to pay.

    The closer you want to get to the desired function [curve] of impedance
    versus
    frequency, the more the cost [cost = total number of R-L-C elements in the
    design].

    Basically what you are trying to doe is very well known in the network
    synthesis
    literature as driving point impedance [DPI] synthesis. [e.g. Darlington's
    method and other similar techniques. Darlngton's technique approaches
    the problem of DPI as the synthesis of a lossless two port terminated
    in an appropriate single resistance.]

    Network synthesis was widely researched, studied and taught back in the
    1940 - 1970 era but... today it is seldom seen, used, or taught. There are
    however lots of older textbooks which cover this field in great depth.

    I'll post a few such references here below for your reference.

    Before you can actually perform the DPI synthesis you will first have to
    find an appropriate rational polynomial function, to form the basis for your
    synthesis, which approximates the impedance function [curve] you desire to
    match. To obtain such a rational polynomial you will have to solve an
    appropriate approximation problem.

    Approximation theory and the techniques for doing this with rational
    polynomial
    are a whole 'nother problem, and other than a few simple graphical straight
    line
    segment tricks, will usually require the use of a computer with an
    appropriate
    algorithm, such as Remez second method, which you may have to write
    yourself!

    Unless you can find consultant to help you... :)

    Check out the following classic texts on network synthesis for a complete
    run down on what you need to do to accomplish your objective:

    1.) Ernst A. Guillemin, "Synthesis of Passive Networks", John Wiley & Sons,
    NY, 1957. [LC# 57-8886. On technical library shelves at LCShelf
    Call # TK3226.G84. See Chapters 3, 4, 9, 10 which cover the DPI synthesis
    in detail, and Chapter 14 which covers the approximation problem.]

    2.) Norman Balabanian, "Network Syntheis" Prentice-Hall, Englewood
    Cliffs, NJ 1958. [LC# 58-11650. On technical library shelves at LCC
    Shelf Call # TK3226.B26. See Chapters 2 & 3 for DPI and Chapter
    9 for the approximation problem.]

    3.) Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill,
    New York, 1962. [LC# 61-16969. On technical library shelves at
    LC Shelf Call # TK3226.W395. See Chapter's 9 & 10 for DPI
    synthesis and Chapter 11 for the approximation problem]

    One does not have to realize such designs with purely passive RLC
    networks and, in appropriate frequency ranges, they can often be
    synthesized with active RC networks [R, C and Op-Amps] by
    appropriate transformations of the passive synthesis results.

    See for instance...

    4.) Adel S. Sedra and Peter O. Brackett, "Filter Theory and Design:
    Active and Passive", Matrix Publishers, Portland, OR, 1978. [LC #
    76-39745. On technical library shelves at LCC Shelf Call #
    TK7872.F5S42.]

    Also, and I have done this myself a couple of times for special low
    frequency
    applications, one can match the analog driving point impedance through
    an appropriate Op-Amp reflectometer circuit to a combination analog
    to digital A/D and digital to analog converter D/A and perform/emulate
    the DPI synthesis in real time using digital signal procssing [DSP]
    techniques. Basically to use the A/D - D/A digital technique to emulate
    the desired DPI you will have to solve the same synthesis and approximation
    problems mentioned above but under a suitable *warping* of the real
    frequency
    axis.

    Hope that all helps... and good luck

    :)
     
  13. Jim Thompson

    Jim Thompson Guest

    On Sun, 22 Feb 2004 22:34:02 GMT, "Peter O. Brackett"

    [snip]
    [snip]

    Gawwwwd! You list makes me feel really old. Not only do I recognize
    all your references, I knew "Ernie" personally... but I had Harry B.
    Lee as my instructor, since I was in VI-B.

    ...Jim Thompson
     
  14. Jim:

    [snip]
    [snip]

    Heh, heh... No one could "splain" driving point synthesis like "Ernie".

    But... Time moves on...

    Y. W. Lee, I have met, Harry B. Lee?

    I never had the pleasure.

    Best Regards,

    --
    Peter K1PO
    Consultant - Signal Processing and Analog Electronics
    Indialantic By-the-Sea, FL.


     
  15. Jim Thompson

    Jim Thompson Guest

    Think War of Independence, this guy was a descendent of "Light-Horse"
    Harry Lee ;-)

    Harry B. Lee was, IMNSHO, better than Ernie. Everything Ernie did was
    1H, 1F and 1 ohm... Harry was a realist, and I still have my notes...
    I don't think anyone could have taught nodal and loop analysis better;
    which is why I attack IC design in that fashion.

    ...Jim Thompson
     
  16. Jim:

    [snip]
    [snip]

    And of course... "pliers" and "soldering iron"...

    [snip]
    Harry was a realist, and I still have my notes...
    [snip]

    As I said, I never had the pleasure, but I'm sure he was a grea teacher...

    [snip]
    [snip]

    Hmmm... how about the "Method of False Assumption"?

    :)
     
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